Electrostatics

Consider an electric dipole comprising two charges $+q$ and $-q$ each with mass $m$, separated by a fixed distance $d$ and initially at rest with its dipole moment pointing along $\hat{\imath}$. A uniform electric field $E \hat{\jmath}$ is ...

Two charges $Q_1 = q$ and $Q_2 = mq$ are placed at the points $P_1(a, b)$ and $P_2(ma, mb)$, respectively, in the $XY$ plane, where $a, b \neq 0$ and $m \neq 0, 1$. If $V_1$ is the potential at a point in the $XY$ plane due to charge $Q_1$ ...

Six infinitely large and thin non-conducting sheets are fixed in configurations I and II. As shown in the figure, the sheets carry uniform surface charge densities which are indicated in terms of $\sigma_0$. The separation between any two c...

A positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and su...

Two co-axial conducting cylinders of same length $\ell$ with radii $\sqrt{2}R$ and $2R$ are kept, as shown in Fig. 1. The charge on the inner cylinder is $Q$ and the outer cylinder is grounded. The annular region between the cylinders is fi...

List-I shows four configurations, each consisting of a pair of ideal electric dipoles. Each dipole has a dipole moment of magnitude $p$, oriented as marked by arrows in the figures. In all the configurations the dipoles are fixed such that ...

An infinitely long thin wire, having a uniform charge density per unit length of $5 \mathrm{nC} / \mathrm{m}$, is passing through a spherical shell of radius $1 \mathrm{~m}$, as shown in the figure. A $10 \mathrm{nC}$ charge is distributed ...

A charge is kept at the central point $\mathrm{P}$ of a cylindrical region. The two edges subtend a half-angle $\theta$ at $\mathrm{P}$, as shown in the figure. When $\theta=30^{\circ}$, then the electric flux through the curved surface of ...

A small electric dipole $\vec{p}_0$, having a moment of inertia $I$ about its center, is kept at a distance $r$ from the center of a spherical shell of radius $R$. The surface charge density $\sigma$ is uniformly distributed on the spherica...

Two beads, each with charge $q$ and mass $m$, are on a horizontal, frictionless, non-conducting, circular hoop of radius $a$. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equ...

An electric dipole is formed by two charges $+q$ and $-q$ located in $x y$-plane at $(0,2) \mathrm{mm}$ and $(0,-2) \mathrm{mm}$, respectively, as shown in the figure. The electric potential at point $P(100,100) \mathrm{mm}$ due to the dipo...

A disk of radius $\mathrm{R}$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is

$$
V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2...

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_{A}=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_{A}=k r$, where $k$ is positive. In the spheric...

A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is $\frac{n q}{6 \epsilon_...

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the...

Two point charges \(-\)Q and +Q/\(\sqrt 3\) are placed in the xy-plane at the origin (0, 0) and a point (2, 0), respectively, as shown in the figure. This results in an equipotential circle of radius R and potential V = 0 in the xy-plane w...

Two point charges \(-\)Q and +Q/\(\sqrt 3\) are placed in the xy-plane at the origin (0, 0) and a point (2, 0), respectively, as shown in the figure. This results in an equipotential circle of radius R and potential V = 0 in the xy-plane w...

A point charge q of mass m is suspended vertically by a string of length l. A point dipole of dipole
moment \(\overrightarrow p\) is now brought towards q from infinity so that the charge moves away. The final
equilibrium position of the s...

Two large circular discs separated by a distance of 0.01 m are connected to a battery via a switch as
shown in the figure. Charged oil drops of density 900 kg m−3
are released through a tiny hole at the
center of the top disc. Once some oil...

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is \(\alpha\). The spheres are...

A circular disc of radius R carries surface charge density
\(\sigma \left( r \right) = {\sigma _0}\left( {1 - {r \over R}} \right)\), where \(\sigma\)0 is a constant
and r is the distance from the center of the disc. Electric flux through ...

One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0, 0).
A point particle of mass m carrying a positive charge q is attached at its other end. The entire system
is kept on a smooth horizontal...

A uniform electric field, \(\overrightarrow E = - 400\sqrt 3 \widehat y\) NC−1
is applied in a region. A charged particle of mass m
carrying positive charge q is projected in this region with an initial speed of 2\(\sqrt {10}\) $$ \times...

An electric dipole with dipole moment \({{{p_0}} \over {\sqrt 2 }}(\widehat i + \widehat j)\) is held fixed at the origin O in the presence of a uniform electric field of magnitude E0.If the potential is constant on a circle of radius R cen...

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V0. A hole with a small area \(\alpha\)4\(\pi\)R2(\(\alpha\) << 1) is made on the shell without affecting the ...

A charged shell of radius R carries a total charge Q. Given \(\phi\) as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a ...

An infinitely long thin non-conducting wire is parallel to the \(z\)-axis and carries a uniform line charge density \(\lambda .\) It pierces a thin non-conducting spherical shell of radius \(R\) in such a way that the arc \(PQ\) subtends an...

A particle, of mass \({10^{ - 3}}\) \(kg\) and charge \(1.0\) \(C,\) is initially at rest. At time \(t=0,\) the particle comes under the influence of an electric field \(\overrightarrow E \left( t \right) = {E_0}\sin \,\,\) $$\omega t\wideh...

The electric field \(E\) is measured at a point \(P(0,0,d)\) generated due to various charge distributions and the dependence of \(E\) on \(d\) is found to be different for different charge distributions. List-\({\rm I}\) contains differen...

A point charge \(+Q\) is placed just outside an imaginary hemispherical surface of radius \(R\) as shown in the figure. Which of the following statements is/are correct?

Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a...

Consider an evacuated cylindrical chamber of height h having rigid conducting plates at the ends and an insulating curved surface as shown in the figure. A number of spherical balls made of a light weight and soft material and coated with a...

Consider a uniform spherical charge distribution of radius \({R_1}\) centred at the origin \(O.\) In this distribution, a spherical cavity of radius \({R_2},\) centred at \(P\) with distance \(OP=a\) \(= {R_1} - {R_2}\) (see figure) is mad...

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density \(\lambda\) are kept parallel to each other. In their resulting electric field, point charges \(q\) and $$-q$...

An infinitely long uniform line charge distribution of charge per unit length \(\lambda\) lies parallel to the y-axis in the y-z plane at \(z = {{\sqrt 3 } \over 2}\)a (see figure). If the magnitude of the flux of the electric field through...

Charges \(Q,\) \(2Q\) and \(4Q\) are uniformly distributed in three dielectric solid spheres \(1,2\) and \(3\) of radii \(R/2,R\) and \(2R\) respectively, as shown in figure. If magnitude of the electric fields at point \(P\) at a distance ...

Four charges Q1, Q2, Q3 and Q4 of same magnitude are fixed along the x axis at x = \(-\)2a, \(-\)a, +a and +2a, respectively. A positive charge q is placed on the positive y axis at a distance b > 0. Four options of the signs of these charg...

Let \({E_1}\left( r \right),{E_2}\left( r \right)\) and \({E_3}\left( r \right)\) be the respective electric field at a distance \(r\) from a point charge \(Q,\) an infinitely long wire with constant linear charge density \(\lambda ,\) and ...

Two non-conducting spheres of radii \({R_1}\) and \({R_2}\) and carrying uniform volume charge densities \(+ \rho\) and \(- \rho ,\) respectively, are placed such that they partially overlap, as shown in the figure. At all points in the ...

Two non-conducting solid spheres of radii \(R\) and \(2R,\) having uniform volume charge densities \({\rho _1}\) and \({\rho _2}\) respectively, touch each other. The net electric field at a distance \(2\) \(R\) from the center of the small...

Six point charges are kept at the vertices of a regular hexagon of side \(L\) and center \(O,\) as shown in the figure. Given that \(K = {1 \over {4\pi {\varepsilon _0}}}{q \over {{L^2}}},\) which of the following statement(s) is (are) corr...

Two large vertical and parallel metal plates having a separation of \(1\) \(cm\) are connected to a \(DC\) voltage source of potential difference \(X\). A proton is released at rest midway between the two plates. It is found to move at $${4...

Consider a thin spherical shell of radius \(R\) with center at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field \(\left| {\overrightarrow E \left( r \right)} \right|\) and th...

A cubical region of side \(a\) has its center at the origin. It encloses three fixed point charges, \(-q\) at \(\left( {0, - a/4,0} \right), + 3q\) at \(\left( {0,0,0} \right)\) and \(-q\) at \(\left( {0, + a/4,0} \right).\) Choose the corr...

An infinitely long solid cylinder of radius R has a uniform volume charge density \(\rho\). It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at ...

Which of the field patterns given below is valid for electric field as well as for magnetic field ?

Which of the following statement(s) is/are correct?

A spherical metal shell A of radius \({R_A}\) and a solid metal sphere \(B\) of radius \({R_B}\left( { < {R_A}} \right)\) are kept far apart and each is given charge \(' + Q'.\) Now they are connected by a thin metal wire. Then

Consider an electric field \(\overrightarrow E = {E_0}\widehat x\) where \({E_0}\) is a constant. The flux through the shaded area (as shown in the figure) due to this field is

Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side a. The surface tension of the soap film is \(\gamma\). The system of charges and planar film are in equilibrium, and $$a = k{\left[ {...